Question:
Solve: $\cos \left\{2 \sin ^{-1}(-x)\right\}=0$
Solution:
Given,
$\cos \left\{2 \sin ^{-1}(-x)\right\}=0$
$\Rightarrow \cos \left(-2 \sin ^{-1} x\right)=0$ $\left[\because \sin ^{-1}(-\theta)=-\sin ^{-1} \theta\right]$
$\Rightarrow \cos \left(2 \sin ^{-1} x\right)=0$ $[\because \cos (-\theta)=\cos \theta]$
We know, $-\frac{\pi}{2} \leq \sin ^{-1} \theta \leq \frac{\pi}{2}$
Therefore, $2 \sin ^{-1} x=\pm \frac{\pi}{2}$
$\Rightarrow \sin ^{-1} x=\pm \frac{\pi}{4}$
$\Rightarrow x=\pm \frac{1}{\sqrt{2}}$
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